trans_isAlgebraic — Mathlib

English

If r ∈ A and f ∈ K[X], Transcendental K (AlgHom.funLike.coe (Polynomial.aeval r) f) is equivalent to Transcendental K r ∧ Transcendental K f.

Русский

Если r ∈ A и f ∈ K[X], транcцендентность Transcendental K (aeval r f) эквивалентна транcцендентности r и f над K.

LaTeX

$$$Transcendental\ K\ (AlgHom.funLike.coe\ (Polynomial.aeval\ r)\ f) \iff (Transcendental\ K\ r \wedge Transcendental\ K\ f)$$$

Lean4

theorem _root_.IsIntegral.trans_isAlgebraic [alg : Algebra.IsAlgebraic R S] {a : A} (h : IsIntegral S a) :
    IsAlgebraic R a := by
  cases subsingleton_or_nontrivial A
  · have := Algebra.IsAlgebraic.nontrivial R S
    exact Subsingleton.elim a 0 ▸ isAlgebraic_zero
  · have := Module.nontrivial S A
    exact h.isAlgebraic.restrictScalars _

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